2001/11/19 by D. Borthwick, David Borthwick, Christopher M. Judge +6 · 1 citation
Mathematics · Physics and Astronomy · #35P25 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.DG #math.SP #msc:35P25 #msc:58J50
paper · pdf · doi:10.48550/arxiv.math/0111211
33 pages, AMS-LaTeX
arxiv created 2001/11/19 · openalex publication_date 2001/11/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.